Method of estimating the lifetime of thermal barrier coatings

ABSTRACT

A method of estimating the lifetime of a thermal barrier coating ( 14 ), which is applied to the surface of a member subjected to cyclical thermal loads, especially a vane and/or blade of a gas turbine, by means of a bond coat ( 12 ) lying in between, leads to more accurate results with simplified calculation by determining, in a first step, the amplitude of the normal stress (Δσ n ) perpendicular to the interface between the bond coat ( 12 ) and the thermal barrier coating ( 14 ) during cyclical loading, and calculating, in a second step, the number N i  of cycles to failure for every normal stress amplitude (Δσ n ) in accordance with the formula  
         N   i     =       (     C        (       Δσ   n       σ   0       )       )     m                   
 
     where σ 0  is a stress reference value and C(δ ox ) and m(δ ox ) are material parameters, which depend on the thickness (δ ox ) of an oxide layer ( 13 ), which is located between the thermal barrier layer ( 14 ) and the bond coat ( 12 ) and grows with the cyclical loading.

TECHNICAL FIELD

[0001] The present invention relates to the technical field of thermallyhighly stressed members, such as for example components (pistons or thelike) of internal combustion engines, gas turbine blades, combustionchambers etc.

PRIOR ART

[0002] Thermally highly stressed components from the high-temperatureregion of gas turbines, such as for example blades or vanes, are coatedfor two reasons:

[0003] to protect the blade or vane material from corrosive attacks, and

[0004] to reduce the temperatures of the metal to a level, which can bewithstood.

[0005] Usually, two coatings are applied to the base material. The firstone is known as the “overlay” coating, which protects against corrosion.The second coating, which is usually referred to as the thermal barriercoating (TBC) and is only applied if need be, serves as theaforementioned thermal isolation.

[0006] However, components coated in such a way may also becomedefective for various reasons: mismatches with respect to thecoefficients of thermal expansion between the various materials of whichthe member consists can cause thermal stresses and deformations in thesystem. Furthermore, instances of scaling due to oxidation and growththereof at relatively high temperatures can produce additional stressloads.

[0007] These stresses and deformations can finally lead to cracking andspalling of the coating layers. According to FIG. 5, the coating system10 is therefore usually made up of four different layers:

[0008] the base material 11, which is usually several mm thick,

[0009] the bond coat 12, which has a thickness of about 0.1 . . . 0.3mm,

[0010] a thermally grown oxide (TGO) layer 13, which grows to athickness of 0.02 mm, and

[0011] the thermal barrier coating 14, which is approximately 0.2 . . .0.4 mm thick.

[0012] In-depth investigations of the stress/deformation behavior havealready been carried out on the basis of finite-element networks, inwhich all four elements of the coating system were modeled in alldetails, including a nonlinear material behavior. It has been found inthese investigations that both the thermal growth of the oxide layersand the creep behavior of the bond coats play a major part in theformation of defects (see in this respect, for example, the article byFreborg, A. M., Ferguson, B. L.: ‘Modelling oxidation induced stressesin thermal barrier coatings’. Material Science and Engineering A245,1998, pages 182 190). However, the results of these investigationscannot be used for predicting lifetimes.

SUMMARY OF THE INVENTION

[0013] It is therefore the object of the invention to specify asimplified method of estimating the lifetime of a thermal barriercoating, which also takes into account the part played by the changingoxide layer.

[0014] The object is achieved by the entirety of the features ofclaim 1. The essence of the invention is to use in the calculation ofthe number N_(i) of cycles to failure material parameters C(δ_(ox)) andm(δ_(ox)) which depend on the thickness (δ_(ox)) of an oxide layer,which is located between the thermal barrier coating 14 and the bondcoat 12 and grows with cyclical loading.

[0015] The calculation is particularly simple here if, according to apreferred refinement of the method according to the invention, thedependence of the material parameters C(δ_(ox)) and m(δ_(ox)) on thethickness (δ_(ox)) of the oxide layer is assumed to be linear, iffurthermore a growth law of the form

δ_(ox) =k _(p) t ^(n)

[0016] with a growth constant kp and an exponent n is used for theincrease in the thickness (δ_(ox)) of the oxide layer with time t, if adamage increment ΔD, which satisfies the approximation formula${\Delta \quad {D(N)}} \approx {\frac{1}{C(N)}\left( {\Delta\sigma}_{n} \right)^{- {m{(N)}}}}$

[0017] is calculated, N giving the number of loading cycles, and C(N)and m(N) being parameters which satisfy the equations

C(N)=α_(c)(NT)^(n)+β_(c)

[0018] and

m(N)=α_(m)(NT)^(n)+β_(m)

[0019] with the exponent n, the constants α_(c), α_(m), β_(c), β_(m),and the holding time T at high temperature per loading cycle, and if thenumber of loading cycles to failure N_(i) of the member is determined bythe damage increment being summed up in accordance with the formula$D = {\sum\limits_{N = 1}^{N_{i}}{\Delta \quad {D(N)}}}$

[0020] until D has reached the value 1.

[0021] Further embodiments are disclosed in the dependent claims.

BRIEF EXPLANATION OF THE FIGURES

[0022] The invention is to be explained in more detail below on thebasis of exemplary embodiments in connection with the drawing, in which

[0023]FIG. 1 shows the simplified, linear dependence of the defectconstant C on the thickness δ_(ox) of the oxide layer assumed for thedetermination of the lifetime;

[0024]FIG. 2 shows the stress components (normal stress and tangentialstress) in the case of a cylindrical test piece;

[0025]FIG. 3 shows experimental results for the lifetime and thethickness of the oxide layer in dependence on the preoxidation period;

[0026]FIG. 4 shows the measured dependence (curve a) and the calculateddependence (curve b) of the number of cycles to failure of the normalstress for a test piece with a 0.6 mm Coatec thermal barrier coating;and

[0027]FIG. 5 shows the schematic structure of a coating system with athermal barrier coating.

WAYS OF IMPLEMENTING THE INVENTION

[0028] Within the scope of the invention, the defect behavior caused bythermocyclic fatigue of the coating system is considered. This approachis justified in a certain way by the results obtained in experiments onvanes and blades in a fluidized bed, from which it becomes evident thata defect begins in the coating at the leading edge of the blade or vaneairfoil and then spreads over the low-pressure side of the test piece asthe number of loading cycles increases.

[0029] Possible fundamental variables which could be significant for thecyclical failure of the TBC layer are:

[0030] the region of mechanical deformation in the layer plane,

[0031] the region of the stress perpendicular to the interface betweenthe TBC layer and the bond coat (normal stress),

[0032] the regions of stored mechanical energy.

[0033] It is assumed that the significant variable for the failure ofthe TBC layer is the region of mechanical stress, which is perpendicularto the interface between the TBC layer and the bond coat (normal stressσ_(n)). It is further presupposed that, if the coating system is loadedwith normal stress cycles of constant stress amplitude (Δσ), and theoxide layer does not grow, the cyclical defect behavior of the TBC layercan be described in approximation by a standard defect equation of theManson-Coffin type $\begin{matrix}{N_{i} = {C\left( \frac{\Delta\sigma}{\sigma_{0}} \right)}^{m}} & (1)\end{matrix}$

[0034] where

[0035] N_(i) is the number of cycles to failure (that is for exampleuntil spalling of the TBC layer at a critical point of the blade or vanesurface),

[0036] C and m are material constants which describe the defect behaviorof the material,

[0037] Δσ is the region of normal stress, and

[0038] σ₀ is a stress reference value.

[0039] It should be noted that the stress reference value go isintroduced in order to make the expression in the parentheses ofequation (1) dimensionless. The value of σ₀ is fixed, for example, at 1N/mm²=10⁻⁶ Mpa.

[0040] The aforementioned assumption corresponds to the assumption thatthe number of cycles to failure can be plotted as a straight line in alog (Δσ) against log (N_(i)) diagram, and that it depends on the uppertemperature of the thermal cycles. In the actual implementation of thecalculation method, this temperature dependence is taken into account bya suitable, piecewise linear interpolation.

[0041] There is also experimental evidence to suggest that the cyclicaldefect resistance of the TBC layer depends on the thickness of the oxidelayer δ_(ox) at the beginning, when no oxide layer has formed as yet,there is the greatest defect resistance. However, after the oxide layerhas grown to a critical thickness (of typically 10 . . . 14 μm), the TBClayer spalls, even if there is no significant cyclical loading. Tosimulate this defect behavior, it is assumed that the lifetimeparameters C and m—in addition to the dependence on temperature—decreaselinearly with increasing thickness δ_(ox) of the oxide layer, asrepresented in FIG. 1 by way of example for the parameter C. A furtherassumption is required for the growth of the thickness δ_(ox) of theoxide layer with time. For this, the following, well-known growth law isused:

δ_(ox) =k _(p) t ^(n).  (2)

[0042] The growth of the oxide layer is accordingly described by agrowth constant k_(p) and an exponent n. The exponent n typically has avalue of 0.5; i.e., the growth over time of the oxide layer is assumedto be proportional to the root of the time. However, the value n can inprinciple also assume other values. Furthermore, it is known of thegrowth constant k_(p) that it increases with temperature. It istherefore introduced as a temperature-dependent variable.

[0043] The defect behavior of the TBC layer is simulated by the defectconstants C and m being assigned a dependence on the oxide layerthickness according to FIG. 1. The defect constant C, for example,decreases with increasing oxide layer thickness δ_(ox) linearly to zero.The linear behavior can be completely fixed by prescribing two points“1” and “2” (FIG. 1), to which the corresponding error constants C₁ andC₂ and also δ₁ and δ₂ are assigned. The defect constant C is in this waya function of the oxide layer thickness according to: $\begin{matrix}{{C\left( \delta_{o\quad x} \right)} = {C_{1} + {\frac{C_{2} - C_{1}}{\delta_{2} - \delta_{1}}\left( {\delta_{o\quad x} - \delta_{1}} \right)}}} & (3)\end{matrix}$

[0044] A corresponding relationship is obtained for the defect exponentm: $\begin{matrix}{{m\left( \delta_{o\quad x} \right)} = {m_{1} + {\frac{m_{2} - m_{1}}{\delta_{2} - \delta_{1}}\left( {\delta_{o\quad x} - \delta_{1}} \right)}}} & (4)\end{matrix}$

[0045] If the growth law from equation (2) is then introduced, thefollowing is obtained for equation (3): $\begin{matrix}{{C(t)} = {C_{1} + {\frac{C_{2} - C_{1}}{\delta_{2} - \delta_{1}}{\left( {{k_{p}t^{n}} - \delta_{1}} \right).}}}} & (5)\end{matrix}$

[0046] A corresponding relationship, which is not given here for thesake of simplicity, is obtained for the defect exponent m. The equation(5) can be written as

C(t)=α_(c) t ^(n)+β_(c)  (6)

[0047] with the following abbreviations: $\begin{matrix}{\alpha_{c} = {\frac{C_{2} - C_{1}}{\delta_{2} - \delta_{1}}k_{p}}} & (7)\end{matrix}$

[0048] and $\begin{matrix}{\beta_{c} = {C_{1} + {\frac{C_{2} - C_{1}}{\delta_{2} - \delta_{1}}\delta_{1}}}} & (8)\end{matrix}$

[0049] The time t is represented in approximation by the product fromthe respective number N of loading cycles and the holding time T (athigh temperature), the holding time being assumed to be long incomparison with the transitional times between the cycles:

t=NT  (9)

[0050] Following from this for (6) is:

C(N)=α_(c)(NT)^(n)+β_(c)  (10)

[0051] and, correspondingly, for the exponent m:

m(N)=α_(m)(NT)^(n)+β_(m)  (11)

[0052] Corresponding then to the defect equation or damage equation (1)is the following equation for the damage rate dD as a function of therate dN of the loading cycles: $\begin{matrix}{{d\quad D} = {{\frac{1}{C(N)}\left( {\Delta\sigma}_{n} \right)^{- {m{(N)}}}d\quad N} = {\frac{1}{{\alpha_{c}\left( {N\quad T} \right)}^{n} + \beta_{c}}\left( {\Delta\sigma}_{n} \right)^{- {({m_{1} + {\alpha_{m}{({N\quad T})}}^{n} + \beta_{m}})}}d\quad {N.}}}} & (12)\end{matrix}$

[0053] A defect occurs if the following conditions apply:

D=1 and N=N _(i)  (13)

[0054] Since the equation (12) cannot be analytically integrated, asimple summation is used. For this purpose, the following approximationis made: $\begin{matrix}{{{\Delta \quad D} \approx {dD}} = {\frac{1}{C(N)}\left( {\Delta\sigma}_{n} \right)^{- {m{(N)}}}}} & (14)\end{matrix}$

[0055] and summed up: $\begin{matrix}{D = {\sum\limits_{N = 1}^{N_{1}}{\Delta \quad {D.}}}} & (15)\end{matrix}$

[0056] Since N, is not known, the summation is continued until one ofthe two conditions is satisfied:

D≧0 or ΔD≦0.  (16)

[0057] As already mentioned, that mechanical variable which is relevantfor the lifetime of the TBC layer is the stress perpendicular to theinterface between the TBC layer and the adjacent bond coat. However,this normal stress cannot be taken from the currently used finiteelement models, because the base material is normally modeled by finitevolume elements. The relatively weak mechanical behavior of the thin topcoats is ignored in this.

[0058] The thickness of the TBC layer is small in comparison with thethickness of the neighboring metal wall. Furthermore, the elasticrigidity of the TBC layer is typically very much less (usually by anorder of magnitude) than the rigidity of the metal. This permits theassumption that the total stress which acts in the TBC layertangentially to the surface of the member is determined by the totaltangential stress of the base material.

[0059] Furthermore, the stress acting perpendicular to the interfacebetween the TBC layer and the bond coat (normal stress) σ_(n), can thenbe estimated if the size of the surface curvature is known. In the caseof a cylindrical member, this estimate corresponds to the known“cylinder formula” $\begin{matrix}{\sigma_{n} = {\frac{t^{\prime}}{R}\sigma_{t}}} & (17)\end{matrix}$

[0060] where σ_(n) represents the normal stress and at represents thetangential stress in the TBC layer and t′ and R denote the thickness ofthe TBC layer and the radius of curvature (see FIG. 2).

[0061] The method actually used for determining the normal stress is alittle more complicated for the following reasons:

[0062] In most cases, there is a two-dimensional tangential stress.

[0063] The members for consideration generally have doubly (convexly,concavely) curved surfaces.

[0064] The determination of the curvature parameters should beapplicable to volume element networks with different types of volumeelements, including linear elements, the surface of which does not haveany independent curvature at all.

[0065] For the estimate of the normal stress, a method which reverts toformulations of the standard shell theory, which are not to be explainedin any more detail here, is therefore used.

[0066] The experience acquired so far on the deformation behavior of theTBC layers suggests the presumption that the stresses in the TBC layerdecrease at high temperatures and therefore disappear during the holdingtimes at high temperatures (i.e. if the member is operated at fullload). After cooling to room temperature, perpendicular tensile stressesdevelop at the interface between the TBC layer and the bond coat,because, in the case of the members (components) concerned, the thermalcoefficients of expansion for the TBC layer are less than for the basematerial.

[0067] It can therefore be assumed that, at room temperature, thecoating system is subjected to perpendicular tensile stress. At hightemperatures, the coating does not undergo any mechanical loading, butthe materials are exposed to corrosive attacks and associateddegradation processes. A nonlinear finite element analysis for multiplecycles, which also takes into account creep effects can predict thecorrect cycle properties here.

[0068] If, however, a linear finite element analysis is used as a basisfor the estimation of the lifetime of the TBC layer, a loading cyclewith reversed characteristics is calculated: no stresses (including theperpendicular stresses) at room temperature and maximum perpendicular(compressive) stresses at operating temperature.

[0069] To overcome these differences and keep the method ofdetermination simple, the absolute values of the regions of the normalstress are used for the determination. However, this approximationneglects possible differences in lifetime for TBC layers with convex andconcave curvature, because the corresponding difference in the algebraicsigns of the perpendicular stresses is ignored.

[0070] The damage coefficient C and the exponent m are fixed byprescribing the values of C₁(T₁) and C₂(T_(i)) for a suitable set ofdifferent temperatures T_(i). Furthermore, the corresponding thicknessvalues δ₁(T_(i)) and δ₂(T_(i)) must be defined.

[0071] C₁(T_(i)) will consequently usually be the temperature-dependentdefect coefficient for the “virginal” TBC system, which has not yetoxidized. In a corresponding way, δ₂(T_(i)) is usually the oxide layerthickness, which immediately makes the TBC layer defective without itbeing subjected to any appreciable fatigue loading.

[0072] To be able to define the parameters of the oxidation rate, valuesfor the exponent m(T₁) and the coefficient k_(p)(T_(i)) of the oxidationrate are required. Finally, the holding time of the loading cyclesconsidered is also required.

[0073] So far, only relatively few experimental data exist for thedetermination of the aforementioned parameters of the model used.However, a series of thermal shock experiments have already been carriedout in a fluidized bed setup, a set of preoxidized blade portions beingused to investigate the spalling behavior of the TBC layers.

[0074] After the preoxidation at three different temperatures, the testpieces were exposed to alternating thermal loading in the fluidized bedinstallation. The holding time at the upper temperature was 5 minutes.The total loading time during the alternating loading in the fluidizedbed was therefore small in comparison with the duration of thepreoxidation. The test results can therefore be used for the roughestimate of the required defect parameters, on the assumption that nochange in the defect parameters C and m takes place during thealternating loading. However, it must be noted that, during theoperation of an actual machine (gas turbine etc.), the holding times areto be measured in hours or even in days, and that usually nopreoxidation takes place there. Therefore, for example in the case of agas turbine blade in use, the defect parameters will most probablychange considerably during the lifetime of the component.

[0075] Some important results of these experimental investigations arereproduced in FIG. 3, where the thickness of the oxide layer is plottedagainst the root of the preoxidation period. In addition, the measurednumbers of load cycles to failure of the TBC layer are entered in thefigure. For many of the experiments, the alternating loading wasdiscontinued after 1000 cycles, so that only a few data points can beused for the estimate of the defect parameters.

[0076] The spalling of the TBC layers usually begins at the leading edgeof the test pieces, where the normal stress is greatest. In acorresponding way, the numbers of load cycles, which led to a firstdefect in the TBC layer (at the leading edge) are entered in FIG. 3. Forall the test pieces except one, however, no further load cycles werecarried out to measure the progression of the layer detachment in theless strongly curved regions of the test pieces.

[0077] The following results can be summarized from FIG. 3 and theinvestigation of the test pieces:

[0078] In the plotting of FIG. 3, the oxide layer thickness isapproximated by straight lines, the slopes of which increase with thetemperature. This suggests that the thickness of the oxide layer isapproximately proportional to the root of the oxidation period.Therefore, the exponent n=0.5 is used for the equation (2). Furthermore,the temperature-dependent values for the coefficient k_(p) can be readoff from the diagram.

[0079] The cyclical properties of the defects in the TBC layer areconfirmed by FIG. 3; the dependence of the numbers of load cycles on thethickness of the oxide layer is obvious.

[0080] The number of load cycles until there is a defect at the frontedge (leading edge) of the blade tends toward zero if the thickness ofthe oxide layer reaches a limit value of 12 . . . 14 μm. It hasadditionally been observed that the defective region becomes larger andspreads from the front edge into less strongly curved regions when thethickness of the oxide layer approaches this limit value. This showsthat, with increasing oxide layer thickness, the defect propertiesbecome more and more independent of the normal stress as soon as theoxide layer thickness approaches the limit value 12 . . . 14 μm.

[0081] Test pieces which have been preoxidized sufficiently long, sothat their oxide layer thickness reached the limit value 12 . . . 14 μm,tend to exhibit spalling over the entire test piece without any furtherload cycles; the TBC layer simply falls from the test piece as soon asit is removed from the preoxidation oven and is cooled to roomtemperature.

[0082] The defect exponent characterizes the dependence of the number ofcycles to failure on the normal stress: an exponent of m=0 means thereis no dependence of the failure on the normal stress. In the case of theaforementioned experiments, this is obviously the case for stronglypreoxidized test pieces, on which the TBC layer spalls from virtuallyall the surface regions irrespective of the curvature. It is thereforeto be expected that the exponent tends toward zero when the thickness ofthe oxide layer approaches its “limit value”.

[0083] To determine the value of the defect exponent m, one of the bladeportions was subjected to continual cyclical loading after theoccurrence of the first defect at the blade tip, while the progressionof the spalled region to less strongly curved regions was recorded. Forthis experiment, a “Coatec” coating with a thickness of 0.6 mm was used.The duration of the preoxidation was 10,000 h at 1000° C. By analogywith the other experiments, a holding time of 5 minutes was used at theupper cycle temperature during the alternating loading in the fluidizedbed installation. Therefore, the total period at the upper temperatureis small in comparison with the preoxidation period, as long as thenumber of cycles is less than about 1000. It can therefore be assumedthat the thickness of the oxide layer is constant throughout thealternating loading.

[0084] As expected, the defect spread into the low-pressure side. On thehigh-pressure side, no defect was observed. The results obtained In thisway are listed in the following table 1: TABLE 1 Progression of theCurvature in Number of defect the defect Normal stress cycles to region[mm] region [mm] [Nmm⁻²] failure 0.0 6.2 11.6 730 50.2 28.9 2.49 86051.8 45.1 1.6 1600 62.8 112.0 0.643 1700 66.0 171.0 0.42 1970

[0085] The tangential stress (thermal mismatch) σ_(t) was estimatedunder the following assumptions:

[0086] thermal coefficient of expansion of the base material:α_(base)=16·10⁻⁶

[0087] thermal coefficient of expansion of the TBC layer:α_(TBC)=10·10⁻⁶

[0088] modulus of elasticity of the TBC layer: E_(TBC)=20,000 Nmm⁻²

[0089] thickness of the TBC layer: 0.6 mm

[0090] Furthermore, the following relationship was used:

σ _(i) =E _(TBC)(α_(base)−α_(TBC))ΔT.  (18)

[0091] The values for the normal stress an were calculated by means ofthe equation (17).

[0092] The resulting numbers of cycles to failure were plotted againstthe normal stress, as represented in FIG. 4 (curve a). An approximationwith the defect parameters m=−0.3 and C=1500 is likewise depicted inFIG. 4 (curve b). These parameters are valid for 1000° C. upper cycletemperature and for alternating loading with an oxide layer with aconstant thickness near the “limit value” of 12 . . . 14 μm. For thecase of thinner oxide layers (i.e. with a shorter preoxidation period),the parameter m will probably assume higher values, perhaps of the orderof magnitude of −2 . . . −5, similar to the m defect parameters of thebase material.

[0093] For the determination of the defect coefficient C, the sameassumptions are made as above, i.e. the thickness of the oxide layer isassumed to be constant during the alternating loading in the fluidizedbed installation.

[0094] For a first rough estimation, it is presupposed that the samedefect exponent m=−0.3 is valid. The defect coefficients C can then becalculated simply according to the equation $\begin{matrix}{C = {\left( \frac{\Delta \quad \sigma}{\sigma_{0}} \right)^{- m}N_{i}}} & (19)\end{matrix}$

[0095] where

[0096] the normal stress Δσ_(n) at the leading or front edge (highestvalue) =11.6 Nmm⁻² and

[0097] σ₀=1.0 Nmm⁻².

[0098] The values obtained in this way for C are listed in the followingtable 2. TABLE 2 Thickness Thickness of the Normal of oxide TemperatureTBC layer stress layer [° C.] [mm] [Nmm⁻²] [μm] Cycles C 1050 0.3 5.66.5 640 380 1050 0.3 5.6 9 160 95 1000 0.3 5.6 10.8 400 240 1000 0.311.1 12 730 350

[0099] The two rows in table 2 for T=1050° C. permit an estimation of Cas a function of the oxide layer thickness, that is for example thedefinition of values C(δ_(ox)) Unfortunately, this is not possible forthe temperature 900° C., because no N_(i) data are available for this.Furthermore, the observed rising C value for the temperature of 1000° C.must be ascribed to a variance in the measured lifetime data.

[0100] For the described calculation method, a computer program with thedesignation COAT has been created, comprising a source code in thelanguage C and using among the input data values which have beencalculated in advance by means of the known finite element programABAQUS.

[0101] Apart from various subprograms within the COAT software, thereare FORTRAN routines which serve as an interface with respect to the.fil file of ABAQUS. Access to the HKS-ABAQUS libraries is notnecessary. The COAT program can therefore run independently of ABAQUS.It is assumed here that the entire data of the mechanical deformation,thermal expansion and temperature are stored in the .fil result file ofABAQUS. Corresponding specifications for this must be prescribed withinthe EL FILE definitions of the ABAQUS inputs. The resulting data for theelements must be stored using the keyword *EL FILE, POSITION=AVERAGED ATNODES. LIST OF REFERENCE NUMERALS 10 coating system 11 base material 12bond coat 13 oxide layer 14 thermal barrier coating (TBC) a, b curve C,C₁, C₂ defect constant δ_(OX), δ₁, δ₂ thickness of oxide layer R radiusof curvature t′ thickness of thermal barrier coating σ_(n) normal stressσ_(t) tangential stress

1. A method of estimating the lifetime of a thermal barrier coating(14), which is applied to the surface of a member subjected to cyclicalthermal loads, especially a vane and/or blade of a gas turbine, by meansof a bond coat (12) lying in between, characterized in that, in a firststep, the amplitude of the normal stress (Δσ_(n)) perpendicular to theinterface between the bond coat (12) and the thermal barrier coating(14) during cyclical loading is determined, and in that, in a secondstep, the number N₁, of cycles to failure is calculated for every normalstress amplitude (Δσ_(n)) in accordance with the formula$N_{i} = \left( {C\left( \frac{\Delta \quad \sigma_{n}}{\sigma_{0}} \right)} \right)^{m}$

where σ₀ is a stress reference value and C(δ_(ox)) and m(δ_(ox)) arematerial parameters, which depend on the thickness (δ_(ox)) of an oxidelayer (13), which is located between the thermal barrier layer (14) andthe bond coat (12) and grows with the cyclical loading.
 2. The method asclaimed in claim 1, characterized in that the dependence of the materialparameters C(δ_(ox)) and m(δ_(ox)) on the thickness (δ_(ox)) of theoxide layer (13) is assumed to be linear.
 3. The method as claimed inclaim 2, characterized in that a growth law of the form δ_(ox) =k _(p) t^(n) with a growth constant k_(p) and an exponent n is used for theincrease in the thickness (δ_(ox)) of the oxide layer (13) with time t,in that a damage increment ΔD, which satisfies the approximation formula${\Delta \quad {D(N)}} \approx {\frac{1}{C(N)}\left( {\Delta \quad \sigma_{n}} \right)^{- {m{(N)}}}}$

is calculated, N giving the number of loading cycles, and C(N) and m(N)being parameters, which satisfy the equations C(N)=α_(c)(NT)^(n)+β_(c)and m(N)=α_(m)(NT)^(n)+β_(m) with the exponent n, the constants α_(c),α_(m), β_(c), β_(m), and the holding time T at high temperature perloading cycle, and in that the number of loading cycles to failure N₁ ofthe member is determined by the damage increment being summed up inaccordance with the formula$D = {\sum\limits_{N = 1}^{Ni}{\Delta \quad {D(N)}}}$

until D has reached the value
 1. 4. The method as claimed in claim 3,characterized in that an exponent n of approximately 0.5 is used in thegrowth law.
 5. The method as claimed in one of claims 1 to 4,characterized in that the normal stress amplitude (Δσ_(n)) on thesurface of the member is determined by means of a finite element method.